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# A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints

by Irina Shevtsova 1,2,3,4,* and Mikhail Tselishchev 2,4,*
1
Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
2
Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, GSP-1, 1-52 Leninskiye Gory, Moscow 119991, Russia
3
Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, Moscow 119333, Russia
4
Moscow Center for Fundamental and Applied Mathematics, Moscow 119991, Russia
*
Authors to whom correspondence should be addressed.
Mathematics 2020, 8(4), 577; https://doi.org/10.3390/math8040577
Received: 18 March 2020 / Revised: 6 April 2020 / Accepted: 8 April 2020 / Published: 13 April 2020
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)
We introduce a generalized stationary renewal distribution (also called the equilibrium transform) for arbitrary distributions with finite nonzero first moment and study its properties. In particular, we prove an optimal moment-type inequality for the Kantorovich distance between a distribution and its equilibrium transform. Using the introduced transform and Stein’s method, we investigate the rate of convergence in the Rényi theorem for the distributions of geometric sums of independent random variables with identical nonzero means and finite second moments without any constraints on their supports. We derive an upper bound for the Kantorovich distance between the normalized geometric random sum and the exponential distribution which has exact order of smallness as the expectation of the geometric number of summands tends to infinity. Moreover, we introduce the so-called asymptotically best constant and present its lower bound yielding the one for the Kantorovich distance under consideration. As a concluding remark, we provide an extension of the obtained estimates of the accuracy of the exponential approximation to non-geometric random sums of independent random variables with non-identical nonzero means. View Full-Text
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Shevtsova, I.; Tselishchev, M. A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints. Mathematics 2020, 8, 577.